Let

${S}_{T}={\mathbb{R}}^{N}\times (0,T)$,

$N\ge 1$ and

$T>0$. The author investigates in

${S}_{T}$ the Cauchy problem for the equation

${u}_{t}=div\left(\right|D{u}^{m}{|}^{p-2}D{u}^{m})+{u}^{q}$ with the initial condition

$u(x,0)={u}_{0}\left(x\right)$. Here

$p>1$,

$m>0$,

$m(p-1)>1$,

$q>1$, and

${u}_{0}$ is locally integrable in

${\mathbb{R}}^{N}$. Of course, for

$m=1$ we have the familiar evolution

$p$-Laplacian equation, and for

$p=2$ we have the porous media equation. The Cauchy problem for the general case is investigated for a large class of initial conditions. To describe this class, the following norm is defined for

$h\ge 1$.

${\left|\right|\left|f\right|\left|\right|}_{h}={sup}_{x\in {\mathbb{R}}^{N}}{\parallel f\parallel}_{h}\left({B}_{1}\left(x\right)\right)$. Here,

${\parallel \xb7\parallel}_{h}$ represents the usual norm in

${L}^{h}\left({B}_{1}\left(x\right)\right)$, where

${B}_{1}\left(x\right)$ denotes the unit ball centered at

$x$ in

${\mathbb{R}}^{N}$. The first result is the following. Assume

${u}_{0}\ge 0$,

$\left|\right||{u}_{0}{\left|\right||}_{h}<\infty $, where

$h=1$ if

$q<m(p-1)+p/N$ and

$h>(N/p)(q-m(p-1\left)\right)$ otherwise. Then there is a constant

${T}_{0}>0$ depending on the data such that a solution

$u(x,t)$ exists in

${S}_{{T}_{0}}$. Quantitative bounds for the solution and results involving a supersolution are obtained. Also the problem of uniqueness is discussed.